Computational and statistical challenges in high dimensional statistical models

Second, we study the computational-statistical gap of the sparse HDLR model; The statistical limit of the model is significantly smaller than its apparent computational limit, which is the minimum signal strength required by known computationally-efficient methods to perform statistical inference. We propose an explanation of the gap by analyzing the Overlap Gap Property (OGP) for HDLR. The OGP is known to be linked with algorithmic hardness in the theory of average-case optimization. We prove that the OGP for HDLR appears, up-to-constants, simultaneously with the computational-statistical gap, suggesting the OGP is a fundamental source of algorithmic hardness for HDLR. Third, we focus on noiseless HDLR. Here we do not assume sparsity, but we make a certain rationality assumption on the coefficients. In this case, we propose a polynomial-time recovery method based on the Lenstra-Lenstra-Lóvasz lattice basis reduction algorithm.